The Lie algebra of the Poincaré group is called the Poincaré algebra and is given by [1]}
\(&\left[P_{i},P_{j}\right] = 0\\&\left[P_{i},P_{0}\right] = 0 \\&\left[J_{i},P_{j}\right] = \epsilon _{ijk}P_{k}\\&\left[J_{i},P_{0}\right] = 0\\&\left[J_{i},J_{j}\right] = \epsilon _{ijk}J_{k}\\&\left[K_{i},P_{j}\right] = \eta _{ij}P_{0}\\&\left[K_{i},P_{0}\right] = P_{i}\\&\left[K_{i},J_{j}\right] = -\epsilon _{ijk}K_{k}\\&\left[K_{i},K_{j}\right] = -\epsilon _{ijk}J_{k}.\) 
